| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find invariant lines through origin |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard linear transformation concepts: computing a determinant (routine), using |det M| for area scaling (direct application of a formula), and verifying an invariant line by showing M maps points on the line back to the line (mechanical substitution). All parts are textbook exercises requiring recall and basic computation rather than problem-solving or insight. |
| Spec | 4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\det(\mathbf{M}) = (4)(-7) - (2)(-5)\) | M1 | Attempt to find \(\det(\mathbf{M})\); calculation alone is sufficient |
| \(\mathbf{M}\) is non-singular because \(\det(\mathbf{M}) = -18\) and so \(\det(\mathbf{M}) \neq 0\) | A1 | Must state \(\det(\mathbf{M}) = -18\) and reference to zero, e.g. \(-18 \neq 0\) and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Area \(R = \dfrac{\text{Area } S}{(\pm) | \det \mathbf{M} | } = \ldots\) |
| \(\text{Area}(R) = \dfrac{63}{ | -18 | } = \dfrac{7}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\begin{pmatrix}4 & -5\\2 & -7\end{pmatrix}\begin{pmatrix}x\\2x\end{pmatrix} = \begin{pmatrix}4x-10x\\2x-14x\end{pmatrix}\) | M1 | Attempts the matrix multiplication shown, or equivalent e.g. \(\begin{pmatrix}\frac{1}{2}y\\y\end{pmatrix}\); may use \(\begin{pmatrix}x\\y\end{pmatrix}\) and substitute \(y=2x\) later |
| \(= \begin{pmatrix}-6x\\-12x\end{pmatrix}\) and so all points on \(y=2x\) map to points on \(y=2x\), hence the line is invariant. OR \(= -6\begin{pmatrix}x\\2x\end{pmatrix}\) hence \(y=2x\) is invariant | A1 | Correct multiplication leading to conclusion that line is invariant; if \(-6\) not extracted, must reference image points being on \(y=2x\) |
# Question 1:
## Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| $\det(\mathbf{M}) = (4)(-7) - (2)(-5)$ | M1 | Attempt to find $\det(\mathbf{M})$; calculation alone is sufficient |
| $\mathbf{M}$ is non-singular because $\det(\mathbf{M}) = -18$ and so $\det(\mathbf{M}) \neq 0$ | A1 | Must state $\det(\mathbf{M}) = -18$ **and** reference to zero, e.g. $-18 \neq 0$ **and** conclusion |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| Area $R = \dfrac{\text{Area } S}{(\pm)|\det \mathbf{M}|} = \ldots$ | M1 | Recalls determinant needed for area scale factor; divides 63 by $\pm$ their determinant |
| $\text{Area}(R) = \dfrac{63}{|-18|} = \dfrac{7}{2}$ | A1ft | $\dfrac{7}{2}$ or follow through $\dfrac{63}{|\text{their det}|}$; must be positive and simplified to single fraction or exact decimal |
## Part (c)
| Working | Mark | Guidance |
|---------|------|----------|
| $\begin{pmatrix}4 & -5\\2 & -7\end{pmatrix}\begin{pmatrix}x\\2x\end{pmatrix} = \begin{pmatrix}4x-10x\\2x-14x\end{pmatrix}$ | M1 | Attempts the matrix multiplication shown, or equivalent e.g. $\begin{pmatrix}\frac{1}{2}y\\y\end{pmatrix}$; may use $\begin{pmatrix}x\\y\end{pmatrix}$ and substitute $y=2x$ later |
| $= \begin{pmatrix}-6x\\-12x\end{pmatrix}$ and so all points on $y=2x$ map to points on $y=2x$, hence the line is invariant. OR $= -6\begin{pmatrix}x\\2x\end{pmatrix}$ hence $y=2x$ is invariant | A1 | Correct multiplication leading to conclusion that line is invariant; if $-6$ not extracted, must reference image points being on $y=2x$ |
---1.
$$\mathbf { M } = \left( \begin{array} { l l }
4 & - 5 \\
2 & - 7
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Show that the matrix $\mathbf { M }$ is non-singular.
The transformation $T$ of the plane is represented by the matrix $\mathbf { M }$.\\
The triangle $R$ is transformed to the triangle $S$ by the transformation $T$.\\
Given that the area of $S$ is 63 square units,
\item find the area of $R$.
\item Show that the line $y = 2 x$ is invariant under the transformation $T$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2019 Q1 [6]}}